I've auditioned both extensively because I was considering the Shures as my next cans (I settled on a used HD600 in the end, though).
Here is my analysis:
Rig: Sony NW-HD3 Line Out
1. All of the Shures are good value for money, with SRH440 delivering the most bang.
2. Unamped, SRH440 sounds really similar to SRH840 in quality. Signature wise, 440 is brighter and livelier and honestly it appeals me more compared to the slower and more laid-back 840. SRH840, unamped, wins in more bass; but not impact. SRH440 has faster decay and the bass, while less than 840, will give a more toe-tapping urge.
SRH440 does NOT lack bass to me. It is neutral, it sounds just nice- real nice. I've heard ATH-M50 and I had ATH-ES7- to me, SRH440 outperforms both even with its lower price point. It produces more detail and neutrality than M50. And it flat-out butchers ES7 in those categories as well as soundstage- 440 is just much nicer to listen to, more natural. ES7 has more bass and an overall more forward, sharp sound signature that will appeal to some people.
3. Out of SPL Phonitor, SRH840 shows its true nature- it retains as much speed as SRH440, with a better bass detail and decay, and a ton more musical. The midrange is very seductive even when compared to SRH440 out of the same amp. Yes, amping the SRH440 doesn't yield as much improvement.
All and all though, I am amazed with what 440 is capable of and I will always recommend them to anyone looking for cans at that price point (below 200 SGD/ 130 USD). At 170 SGD they flat out kill ATH M50 and ES7 (both 200 SGD).
Oh, and keep in mind that I listened to 80% electric guitar based rock and blues. Those really rock with SRH440. When I played tracks such as slow jazz and vocal, unamped SRH840 did do it better than 440- this may weigh against my argument that SRH440 is preferable to SRH840 unamped.